Square root of 5

Square root of 5
	The diagonal of a 2 × 1 rectangle has length √5.
Rationality	Irrational
Representations
Decimal	2.23606797749978969...
Algebraic form
Continued fraction

The square root of 5, denoted ⁠ ${\sqrt {5}}$ ⁠, is the positive real number that, when multiplied by itself, gives the natural number 5. Along with its conjugate ⁠ $-{\sqrt {5}}$ ⁠, it solves the quadratic equation ⁠ $x^{2}-5=0$ ⁠, making it a quadratic integer, a type of algebraic number. ⁠ ${\sqrt {5}}$ ⁠ is an irrational number, meaning it cannot be written as a fraction of integers.^[1] The first forty significant digits of its decimal expansion are:

2.236067977499789696409173668731276235440... (sequence A002163 in the OEIS).

A length of ⁠ ${\sqrt {5}}$ ⁠ can be constructed as the diagonal of a ⁠ $2\times 1$ ⁠ unit rectangle. ⁠ ${\sqrt {5}}$ ⁠ also appears throughout in the metrical geometry of shapes with fivefold symmetry; the ratio between diagonal and side of a regular pentagon is the golden ratio ⁠ $\varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}$ ⁠.

Rational approximations

The square root of 5 is an irrational number, meaning it can not be exactly represented as a fraction ⁠ $x/y$ ⁠ where ⁠ $x$ ⁠ and ⁠ $y$ ⁠ are integers. However, it can be approximated arbitrarily closely by such rational numbers.

Particularly good approximations are the integral solutions of Pell's equations,

x^{2}-5y^{2}=1\quad {\text{and}}\quad x^{2}-5y^{2}=-1,

which can be algebraically rearranged into the form

{\frac {x}{y}}={\sqrt {5\pm {\frac {1}{y^{2}}}}}.

For example, the approximation ⁠ $\textstyle 2={\sqrt {5-1}}$ ⁠, which is accurate to about 10%, satisfies the negative Pell's equation, ⁠ $\textstyle 2^{2}-5\cdot 1^{2}=4-5=-1$ ⁠; likewise, the approximation ⁠ $\textstyle {\tfrac {9}{4}}={\sqrt {5+{\tfrac {1}{16}}}}=2.25$ ⁠, which is accurate within 1%, satisfies the positive equation, ⁠ $\textstyle 9^{2}-5\cdot 4^{2}=81-80=1$ ⁠. These two approximations are the respective fundamental solutions of each Pell's equation, to which additional solutions are algebraically related.

Solutions to both Pell's equations can also be found systematically by following the Euclidean algorithm, resulting in the simple continued fraction for ⁠ ${\sqrt {5}}$ ⁠,^[2]

{\sqrt {5}}=[2;4,4,4,\ldots {}]=2+{\cfrac {1}{4+{\cfrac {1}{4+{\cfrac {1}{4+{{\vphantom {x}} \atop \displaystyle \ddots }}}}}}}.

Each step ⁠ $n$ ⁠ of the algorithm produces a better approximation ⁠ $x_{n}/y_{n}$ ⁠, one of the convergents (partial evaluations) of this continued fraction. These are a sequence of best rational approximations to ⁠ ${\sqrt {5}}$ ⁠, each more accurate than any other rational approximation with the same or smaller denominator. They give all of the solutions to Pell's equations, satisfying ⁠ $\textstyle x_{n}^{2}-5y_{n}^{2}=(-1)^{n}$ ⁠.^[3] The first several convergents to the continued fraction are:^[4]


⁠ ${\boldsymbol {n}}$ ⁠	⁠ $0$ ⁠	⁠ $1$ ⁠	⁠ $2$ ⁠	⁠ $3$ ⁠	⁠ $4$ ⁠	⁠ $5$ ⁠	⁠ $6$ ⁠	⁠ $7$ ⁠	⁠ $8$ ⁠	⁠ $9$ ⁠	⁠ $\ldots$ ⁠
⁠ ${\frac {{\boldsymbol {x_{n}}}{\vphantom {t}}}{\boldsymbol {y_{n}}}}$ ⁠	${\frac {1}{0}}$	${\frac {2}{1}}$	${\frac {9}{4}}$	${\frac {38}{17}}$	${\frac {161}{72}}$	${\frac {682}{305}}$	${\frac {2889}{1292}}$	${\frac {12238}{5473}}$	${\frac {51841}{23184}}$	${\frac {219602}{98209}}$	$\ldots$

In the limit, these approximations converge to ⁠ ${\sqrt {5}}$ ⁠. That is, ⁠ $\textstyle \lim _{n\to \infty }x_{n}/y_{n}={\sqrt {5}}$ ⁠.

One of the oldest methods of calculating a square root of a number ⁠ $d$ ⁠, the Babylonian method, starts with an initial guess ⁠ $x_{0}$ ⁠, and at each step finds a new approximation by averaging the previous approximation and ⁠ $d$ ⁠ times its reciprocal, ⁠ $x_{n+1}={\tfrac {1}{2}}(x_{n}+d/x_{n})$ ⁠. This is the special case, for the function ⁠ $\textstyle f(x)=x^{2}-d$ ⁠, of Newton's method for finding the root of an arbitrary function. For a typical guess, the approximation converges quadratically (roughly doubles the number of correct digits at each step).

The initial guess is somewhat arbitrary, but when approximating ⁠ ${\sqrt {5}}$ ⁠ by this method, usually ⁠ $x_{0}=2$ ⁠ is chosen. With this choice, the ⁠ $n$ ⁠th approximation is equal to the ⁠ $2^{n}$ ⁠th convergent of the continued fraction for ⁠ ${\sqrt {5}}$ ⁠.

{\begin{aligned}x_{0}&={\frac {2}{1}}&\!\!x_{1}&={\frac {9}{4}}&\!\!x_{2}&={\frac {161}{72}}&\!\!x_{3}&={\frac {51841}{23184}}&\!\!\!\!\ldots ,\\[2mu]&=2.{\color {BrickRed}0},&&=2.2{\color {BrickRed}5},&&\approx 2.236{\color {BrickRed}1},&&\approx 2.23606\,7977{\color {BrickRed}9},\end{aligned}}

with digits that differ from those of ⁠ ${\sqrt {5}}$ ⁠ highlighted in red.

Relation to the golden ratio and Fibonacci numbers

The golden ratio ⁠ $\varphi$ ⁠ is the arithmetic mean of 1 and ${\sqrt {5}}$ .^[5] ${\sqrt {5}}$ has a relationship to the golden ratio and its algebraic conjugate ⁠ ${\overline {\varphi }}$ ⁠ as is expressed in the following formulae:

{\begin{aligned}{\sqrt {5}}&=\varphi -{\overline {\varphi }}=2\varphi -1=1-2{\overline {\varphi }},\\[5pt]\varphi &={\frac {1+{\sqrt {5}}}{2}}={\overline {\varphi }}+{\sqrt {5}}=-{\frac {1}{~\!{\overline {\varphi }}\!~}}=1-{\overline {\varphi }},\\[5pt]{\overline {\varphi }}&={\frac {1-{\sqrt {5}}}{2}}=\varphi -{\sqrt {5}}=-{\frac {1}{\varphi }}=1-\varphi .\end{aligned}}

${\sqrt {5}}$ then figures in the closed form expression for the Fibonacci numbers:^{[citation needed]}

F(n)={\frac {\varphi ^{n}-{\overline {\varphi }}^{n}}{\sqrt {5}}}.

The quotient ⁠ ${\sqrt {5}}{\big /}\varphi$ ⁠ provides an interesting pattern of continued fractions and are related to the ratios between the Fibonacci numbers and the Lucas numbers:^[6]

{\begin{aligned}{\frac {\sqrt {5}}{\varphi }}={\frac {5-{\sqrt {5}}}{2}}&=1.381966\dots =[1;2,1,1,1,\ldots ]\end{aligned}}

The convergents feature the Lucas numbers as numerators and the Fibonacci numbers as denominators:

{\frac {1}{1}},{\frac {3}{2}},{\frac {4}{3}},{\frac {7}{5}},{\frac {11}{8}},{\frac {18}{13}},{\frac {29}{21}},\ldots ,{\frac {L_{n}}{F_{n+1}}},\ldots

In the limit,

\lim _{n\to \infty }{\frac {F_{n+1}}{F_{n}}}={\frac {L_{n+1}}{L_{n}}}=\varphi ,\qquad \lim _{n\to \infty }{\frac {L_{n}}{F_{n}}}={\sqrt {5}}.

More precisely, the convergents to the continued fraction for ⁠ ${\sqrt {5}}$ ⁠ (see § Rational approximations above) are:

{\frac {2}{1}},{\frac {9}{4}},{\frac {38}{17}},{\frac {161}{72}},{\frac {682}{305}},\ldots ,{\frac {{\tfrac {1}{2}}L_{3n}}{{\tfrac {1}{2}}F_{3n}}},\ldots .

Geometry

Geometrically, ${\sqrt {5}}$ corresponds to the diagonal of a rectangle whose sides are of length 1 and 2, as is evident from the Pythagorean theorem. Such a rectangle can be obtained by halving a square, or by placing two equal squares side by side. This can be used to subdivide a square grid into a tilted square grid with five times as many squares, forming the basis for a subdivision surface.^[7] Together with the algebraic relationship between ${\sqrt {5}}$ and ⁠ $\varphi$ ⁠, this forms the basis for the geometrical construction of a golden rectangle from a square, and for the construction of a regular pentagon given its side (since the side-to-diagonal ratio in a regular pentagon is ⁠ $\varphi$ ⁠).

Since two adjacent faces of a cube would unfold into a ⁠ $1\mathbin {:} 2$ ⁠ rectangle, the ratio between the length of the cube's edge and the shortest distance from one of its vertices to the opposite one, when traversing the cube surface, is ${\sqrt {5}}$ . By contrast, the shortest distance when traversing through the inside of the cube corresponds to the length of the cube diagonal, which is the square root of three times the edge.^[8]

A rectangle with side proportions $1\mathbin {:} {\sqrt {5}}$ is part of the series of dynamic rectangles, which are based on proportions ⁠ ${\sqrt {1}}$ ⁠, ⁠ ${\sqrt {2}}$ ⁠, ⁠ ${\sqrt {3}}$ ⁠, ⁠ ${\sqrt {4}}$ ⁠, ⁠ ${\sqrt {5}}$ ⁠, ... and successively constructed using the diagonal of the previous root rectangle, starting from a square.^[9] A root-5 rectangle is particularly notable in that it can be split into a square and two equal golden rectangles or into two golden rectangles of different sizes.^[10] It can also be decomposed as the union of two equal golden rectangles whose intersection forms a square. These shapes pictorially represent the algebraic relationships between ${\sqrt {5}}$ , ⁠ $\varphi$ ⁠ and ⁠ $\varphi ^{-1}$ ⁠ mentioned above.

Trigonometry

The square root of 5 appears in trigonometric constants related to the angles in a regular pentagon and decagon, which when combined which can be combined with other angles involving ${\sqrt {2}}$ and ${\sqrt {3}}$ to describe sines and cosines of every angle whose measure in degrees is divisible by 3 but not by 15.^[11] The simplest of these are

{\begin{aligned}\sin {\frac {\pi }{10}}=\sin 18^{\circ }&={\tfrac {1}{4}}({\sqrt {5}}-1)={\frac {1}{{\sqrt {5}}+1}},\\[5pt]\sin {\frac {\pi }{5}}=\sin 36^{\circ }&={\tfrac {1}{4}}{\sqrt {2(5-{\sqrt {5}})}},\\[5pt]\sin {\frac {3\pi }{10}}=\sin 54^{\circ }&={\tfrac {1}{4}}({\sqrt {5}}+1)={\frac {1}{{\sqrt {5}}-1}},\\[5pt]\sin {\frac {2\pi }{5}}=\sin 72^{\circ }&={\tfrac {1}{4}}{\sqrt {2(5+{\sqrt {5}})}}\,.\end{aligned}}

Computing its value was therefore historically important for generating trigonometric tables. Since ${\sqrt {5}}$ is geometrically linked to half-square rectangles and to pentagons, it also appears frequently in formulae for the geometric properties of figures derived from them, such as in the formula for the volume of a dodecahedron.^[8]

Diophantine approximations

Hurwitz's theorem in Diophantine approximations states that every irrational number $x$ can be approximated by infinitely many rational numbers $⁠ m / n ⁠$ in lowest terms in such a way that

\left|x-{\frac {m}{n}}\right|<{\frac {1}{{\sqrt {5}}\,n^{2}}}

and that ${\sqrt {5}}$ is best possible, in the sense that for any larger constant than ${\sqrt {5}}$ , there are some irrational numbers $x$ for which only finitely many such approximations exist.^[12]

Closely related to this is the theorem^[13] that of any three consecutive convergents $⁠ p i / q i ⁠$ , $⁠ p i +1 / q i +1 ⁠$ , $⁠ p i +2 / q i +2 ⁠$ , of a number $α$ , at least one of the three inequalities holds:

\left|\alpha -{p_{i} \over q_{i}}\right|<{1 \over {\sqrt {5}}q_{i}^{2}},\quad \left|\alpha -{p_{i+1} \over q_{i+1}}\right|<{1 \over {\sqrt {5}}q_{i+1}^{2}},\quad \left|\alpha -{p_{i+2} \over q_{i+2}}\right|<{1 \over {\sqrt {5}}q_{i+2}^{2}}.

And the ${\sqrt {5}}$ in the denominator is the best bound possible since the convergents of the golden ratio make the difference on the left-hand side arbitrarily close to the value on the right-hand side. In particular, one cannot obtain a tighter bound by considering sequences of four or more consecutive convergents.^[13]

Algebra

The two quadratic fields ⁠ $\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$ ⁠ and ⁠ $\mathbb {Q} {\bigl (}{\sqrt {-5}}~\!{\bigr )}$ ⁠, field extensions of the rational numbers, and their associated rings of integers, ⁠ $\mathbb {Z} {\bigl [}{\tfrac {1}{2}}+{\tfrac {1}{2}}{\sqrt {5}}~\!]$ ⁠ and ⁠ $\mathbb {Z} {\bigl [}{\sqrt {-5}}~\!]$ ⁠, respectively, are basic examples and have been studied extensively.

The ring $\mathbb {Z} [{\sqrt {-5}}]$ contains numbers of the form $a+b{\sqrt {-5}}$ , where $a$ and $b$ are integers and ${\sqrt {-5}}$ is the imaginary number $i{\sqrt {5}}$ . This ring is a frequently cited example of an integral domain that is not a unique factorization domain.^[14] For example, the number 6 has two inequivalent factorizations within this ring: $6=2\cdot 3=(1-{\sqrt {-5}})(1+{\sqrt {-5}}).\,$

On the other hand, the real quadratic integer ring of golden integers $\mathbb {Z} [\varphi ]$ , adjoining the golden ratio $\varphi ={\tfrac {1}{2}}{\bigl (}1+{\sqrt {5}}~\!{\bigr )}$ , was shown to be Euclidean, and hence a unique factorization domain, by Dedekind. This is the ring of integers in the golden field ⁠ $\mathbb {Q} {\bigl (}{\sqrt {5}}~\!{\bigr )}$ ⁠.^{[citation needed]}

The field $\mathbb {Q} [{\sqrt {-5}}],$ like any other quadratic field, is an abelian extension of the rational numbers. The Kronecker–Weber theorem therefore guarantees that the square root of five can be written as a rational linear combination of roots of unity:

{\sqrt {5}}=e^{{\frac {2\pi }{5}}i}-e^{{\frac {4\pi }{5}}i}-e^{{\frac {6\pi }{5}}i}+e^{{\frac {8\pi }{5}}i}.\,

^{[citation needed]}

Decimal calculation

As of January 2022, the numerical value in decimal of the square root of 5 has been computed to at least 2.25 trillion digits.^[15]

References

^ Dauben, Joseph W. (June 1983) Scientific American Georg Cantor and the origins of transfinite set theory. Volume 248; Page 122.
^ (sequence A040002 in the OEIS)
^ Conrad, Keith. "Pell's Equation II" (PDF). uconn.edu. Retrieved 17 March 2022.
^ Their numerators are 2, 9, 38, 161, … (sequence A001077 in the OEIS).
Their denominators are 1, 4, 17, 72, … (sequence A001076 in the OEIS).
^ Browne, Malcolm W. (July 30, 1985) New York Times Puzzling Crystals Plunge Scientists into Uncertainty. Section: C; Page 1. (Note: this is a widely cited article).
^ Richard K. Guy: "The Strong Law of Small Numbers". American Mathematical Monthly, vol. 95, 1988, pp. 675–712
^ Ivrissimtzis, Ioannis P.; Dodgson, Neil A.; Sabin, Malcolm (2005), " ${\sqrt {5}}$ -subdivision", in Dodgson, Neil A.; Floater, Michael S.; Sabin, Malcolm A. (eds.), Advances in multiresolution for geometric modelling: Papers from the workshop (MINGLE 2003) held in Cambridge, September 9–11, 2003, Mathematics and Visualization, Berlin: Springer, pp. 285–299, doi:10.1007/3-540-26808-1_16, ISBN 3-540-21462-3, MR 2112357
^ ^a ^b Sutton, David (2002). Platonic & Archimedean Solids. Walker & Company. p. 55. ISBN 0802713866.
^ Kimberly Elam (2001), Geometry of Design: Studies in Proportion and Composition, New York: Princeton Architectural Press, ISBN 1-56898-249-6
^ Jay Hambidge (1967), The Elements of Dynamic Symmetry, Courier Dover Publications, ISBN 0-486-21776-0
^ Julian D. A. Wiseman, "Sin and cos in surds"
^ LeVeque, William Judson (1956), Topics in number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass., MR 0080682
^ ^a ^b Khinchin, Aleksandr Yakovlevich (1964), Continued Fractions, University of Chicago Press, Chicago and London
^ Chapman, Scott T.; Gotti, Felix; Gotti, Marly (2019), "How do elements really factor in $\mathbb {Z} [{\sqrt {-5}}]$ ?", in Badawi, Ayman; Coykendall, Jim (eds.), Advances in Commutative Algebra: Dedicated to David F. Anderson, Trends in Mathematics, Singapore: Birkhäuser/Springer, pp. 171–195, arXiv:1711.10842, doi:10.1007/978-981-13-7028-1_9, ISBN 978-981-13-7027-4, MR 3991169, S2CID 119142526, Most undergraduate level abstract algebra texts use $\mathbb {Z} [{\sqrt {-5}}]$ as an example of an integral domain which is not a unique factorization domain
^ Yee, Alexander. "Records Set by y-cruncher".

[1] Dauben, Joseph W. (June 1983) Scientific American Georg Cantor and the origins of transfinite set theory. Volume 248; Page 122.

[2] (sequence A040002 in the OEIS)

[conrad-3] Conrad, Keith. "Pell's Equation II" (PDF). uconn.edu. Retrieved 17 March 2022.

[4] Their numerators are 2, 9, 38, 161, … (sequence A001077 in the OEIS).
Their denominators are 1, 4, 17, 72, … (sequence A001076 in the OEIS).

[5] Browne, Malcolm W. (July 30, 1985) New York Times Puzzling Crystals Plunge Scientists into Uncertainty. Section: C; Page 1. (Note: this is a widely cited article).

[6] Richard K. Guy: "The Strong Law of Small Numbers". American Mathematical Monthly, vol. 95, 1988, pp. 675–712

[7] Ivrissimtzis, Ioannis P.; Dodgson, Neil A.; Sabin, Malcolm (2005), " ${\sqrt {5}}$ -subdivision", in Dodgson, Neil A.; Floater, Michael S.; Sabin, Malcolm A. (eds.), Advances in multiresolution for geometric modelling: Papers from the workshop (MINGLE 2003) held in Cambridge, September 9–11, 2003, Mathematics and Visualization, Berlin: Springer, pp. 285–299, doi:10.1007/3-540-26808-1_16, ISBN 3-540-21462-3, MR 2112357

[:0-8] Sutton, David (2002). Platonic & Archimedean Solids. Walker & Company. p. 55. ISBN 0802713866.

[9] Kimberly Elam (2001), Geometry of Design: Studies in Proportion and Composition, New York: Princeton Architectural Press, ISBN 1-56898-249-6

[10] Jay Hambidge (1967), The Elements of Dynamic Symmetry, Courier Dover Publications, ISBN 0-486-21776-0

[11] Julian D. A. Wiseman, "Sin and cos in surds"

[12] LeVeque, William Judson (1956), Topics in number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass., MR 0080682

[khinchin-13] Khinchin, Aleksandr Yakovlevich (1964), Continued Fractions, University of Chicago Press, Chicago and London

[14] Chapman, Scott T.; Gotti, Felix; Gotti, Marly (2019), "How do elements really factor in $\mathbb {Z} [{\sqrt {-5}}]$ ?", in Badawi, Ayman; Coykendall, Jim (eds.), Advances in Commutative Algebra: Dedicated to David F. Anderson, Trends in Mathematics, Singapore: Birkhäuser/Springer, pp. 171–195, arXiv:1711.10842, doi:10.1007/978-981-13-7028-1_9, ISBN 978-981-13-7027-4, MR 3991169, S2CID 119142526, Most undergraduate level abstract algebra texts use $\mathbb {Z} [{\sqrt {-5}}]$ as an example of an integral domain which is not a unique factorization domain

[15] Yee, Alexander. "Records Set by y-cruncher".

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